Crackin' Good Mathematics

A couple of years ago, Alain Goriely of the University of Arizona visited Hungary for a mathematical conference. He took a break to listen to some local bands and watch entertainers crack bullwhips. Watching a whip snake rapidly through the air and then outplay the bands with a thunderous snap, Goriely wondered what created the crack.

The fictional Zorro may have defended himself with a long whip, but a bullwhip was not designed as a weapon. The bullwhip—really just a long whip—was developed as a way to control a herd of cattle or to signal someone out of yelling distance. It consists of a handle followed by a long, braided section called a thong that tapers down to a fine end, or "cracker." Experts crack bullwhips in a variety of styles. Most pull the whip over the head and back, cracking it far in front of them. Others swing it sidearm. Some swing it their own way.

In the early 1900s, some scientists wondered whether a whip's crack came from a sonic boom. That is, perhaps part of the whip moves faster than the speed of sound, around 750 miles an hour, and the clap of noise comes as the sound barrier is broken. Presumably the cracker creates the crack. By the 1920s, high-speed photography revealed that a whip's cracker can indeed break the sound barrier. The question for Goriely, however, was: How can the relatively slow speed of a whip pulled back and forth generate such high speeds at the tip?

Goriely said, "The first thing I did—after looking at the old papers—was buy a whip." He bought one through the online marketplace eBay for $15. "It was a really crappy whip," Goriely said. Although he bought a book and videotape on bullwhip cracking, he couldn't get a squeak out of his. He moved up to a better whip, one that cost $70, and started cracking it right away. "You realize that you need a good whip to make it work," he said. In fact, someone fancying a decent whip can easily pay a few hundred dollars or more.

More than a good whip, Goriely needed a good teammate. He had one in graduate student Tyler McMillen. McMillen said, "We had been doing problems with elastic rods, so we thought a bullwhip would be an interesting problem." Goriely and McMillen started by examining past approaches. In general, previous investigators selected a law of conservation, made an assumption about the shape of the whip and combined that information to calculate the velocity of the tip. Different conservation models—say, conservation of energy vs. conservation of linear momentum—can give different results: The tip approaches infinite speed in some models and maintains a constant speed in others. McMillen said, "The problem is: You can't assume the shape of the rod, because the rod obeys a physical law, and you must solve for that shape."

This solution demanded cracking and computation. Goriely said, "I was in my backyard cracking, and Tyler did all the hard work." McMillen started with what he called a pretty simple model of an elastic rod, basically a rod that can bend as a whip does. He let the radius of his model vary, so that its shape mimicked a whip's taper, and he modeled the unfurling of the whip as a traveling loop. McMillen said, "We derived the equation for the rod with the varying cross section and looked at how this loop would change as you allowed the cross section to vary. That required some advanced mathematical techniques." Goriely and McMillen found that in fact a loop can stay the same size as it goes down the rod and still accelerate.

The crack happens when the loop reaches the end of the whip and opens. When Goriely and McMillen's model used a tapering that made a virtual whip's tip just one-tenth the diameter of the handle, the tip reached speeds 32 times faster than the original speed of the loop. As a result, this model bullwhip's tip can break the sound barrier rather easily. (In real whips, the increase in speed could be much higher, because McMillen said that real whips often taper even more.) Getting to this answer required conservation of energy, as well as conservation of linear and angular momentum. In other words, Goriely and McMillen created a much more realistic picture of the dynamics of cracking.

This work, however, goes beyond curiosity about the sound of a whip. For one thing, whip cracking provided Goriely with a new teaching tool. He said, "I do take it to show the students because people are very impressed by whip cracking." Moreover, Goriely and McMillen's model provides added insight for the general problem of the motion of waves in complex materials. One day, investigators might use this work to understand the motility of bacteria and sperm or the waves along DNA as it unfolds to make RNA. The role of the bullwhip might eventually stretch far beyond herding dogies--all because cracking caught Goriely's attention.--Mike May